Question: EXERCISE 4.36. In Example 4.20, use Eq. 4.16 to verify that the fake spheres with a = 2 and a = , have no umbilical

EXERCISE 4.36. In Example 4.20, use Eq. 4.16 to verify that the fake spheres with a = 2 and a = , have no umbilical points. (4.16) T. EXAMPLE 4.20 (Fake Spheres). We wish to construct surfaces of revolution with constant Gaussian curvature equal to 1. If K = 1, then Eq. 4.15 says that x" = -x. The following is a solution for every a > 0: (4.17) x(t) = a cos(t). The hypothesis that y is parametrizationgth, so (x')2 + (2')2 = 1, means that z is determined by x as follows: (4.18) = (t) = VI - x'(s)2 ds = 1 - a2 sin? (s) ds. (4.15) K1. Consider the simple regression model: V/i = Po+ Piri +ui, for i = 1, ..., n, with E(ur,) 7 0 and let z be a dummy instrumental variable for a, such that we can write: with E(uilz;) = 0 and E(vilzi) = 0. (c) Denote by no, the number of observations for which z =0 and by n, the number of observations for which z, = 1. Show that: (a - 2) = =(n-m). 1=1 and that: [(8 -=)(3: - 9) = 7 "(n - ni) (31 - 90) . where to and g are the sample means of y for z equal to 0 and 1 respectively. (Hint: Use the fact that n = nj + no, and that = = m). (d) Now we regress y on i to obtain an estimator of 81. From the standard formula of the slope estimator for an OLS regression and using the result in (c), show that: By1 - 90 I1 - To This estimator is called the Wald estimator.1. Consider the simple regression model: yi = Po+ Biz, + ui, for i = 1, ... . n. with E(uilx;) # 0 and let z be a dummy instrumental variable for z, such that we can write: Ti = no + 12+ vi, with E(uilz;) = 0 and E(vilz;) = 0. a) show that no = To, and #1 = 21 - 10, where ro and I, are the sample means of a for 2 equal to 0 and 1 respectively. (b) Define: i; = Fo + (21 - 10) Zi, show that, I, - I = (1, - To) (2, - 2), where ? is the sample mean of z. solve both parts, econometrics8= Bot ByX Assumption (Under simple linear regression model ) I . X2 .... Xn are nonrandom II. X1 .. X, are not all equa III . E ( Us ) =0, E ( U. ) = 0.. E ( un) = 0 IZ. var (us) = var (U2)= var(us) In other words, var (ulx) cdoes not depend on X ( HomesKaclasticty ) V. Us, Uz.. Un are mutually indeperant Provide a mathematical proof for the equation E (62 ) = 62 using assumptions I! - V. 62 = n - 2 n

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